libinsighttoolkit4-dev 4.5.0-3 / usr / include / ITK-4.5 / itkFEMSolverCrankNicolson.hxx

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 *
 *  Copyright Insight Software Consortium
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *         http://www.apache.org/licenses/LICENSE-2.0.txt
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
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 *
 *=========================================================================*/
#ifndef __itkFEMSolverCrankNicolson_hxx
#define __itkFEMSolverCrankNicolson_hxx

#include "itkFEMSolverCrankNicolson.h"

#include "itkFEMLoadNode.h"
#include "itkFEMLoadElementBase.h"
#include "itkFEMLoadBC.h"
#include "itkFEMLoadBCMFC.h"
#include "itkFEMLoadLandmark.h"

namespace itk
{
namespace fem
{
#define TOTE

/**
  * Default constructor which sets the indices for the matrix and vector storage.
  * Time step and other parameters are also initialized.
  */
template <unsigned int VDimension>
SolverCrankNicolson<VDimension>
::SolverCrankNicolson()
{
  m_TimeStep = 0.5;
  m_Rho = 1.;
  m_Alpha = 0.5;
  // BUG FIXME NOT SURE IF SOLVER IS USING VECTOR INDEX 1 FOR BCs
  m_ForceTIndex = 0;                      // vector
  m_ForceTMinus1Index = 2;                // vector
  m_SolutionVectorTMinus1Index = 3;       // vector
  m_DiffMatrixBySolutionTMinus1Index = 4; // vector
  m_ForceTotalIndex = 5;                  // vector
  m_SolutionTIndex = 0;                   // solution
  m_TotalSolutionIndex = 1;               // solution
  m_SolutionTMinus1Index = 2;             // solution
  m_SumMatrixIndex = 0;                   // matrix
  m_DifferenceMatrixIndex = 1;            // matrix
  m_CurrentMaxSolution = 1.0;
  m_UseMassMatrix = true;
  m_Iterations = 0;
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::InitializeForSolution()
{
  this->m_ls->SetSystemOrder(this->m_NGFN + this->m_NMFC);
  this->m_ls->SetNumberOfVectors(6);
  this->m_ls->SetNumberOfSolutions(3);
  this->m_ls->SetNumberOfMatrices(2);
  this->m_ls->InitializeMatrix(m_SumMatrixIndex);
  this->m_ls->InitializeMatrix(m_DifferenceMatrixIndex);
  this->m_ls->InitializeVector(m_ForceTIndex);
  this->m_ls->InitializeVector(m_ForceTotalIndex);
  this->m_ls->InitializeVector(m_ForceTMinus1Index);
  this->m_ls->InitializeVector(m_SolutionVectorTMinus1Index);
  this->m_ls->InitializeVector(m_DiffMatrixBySolutionTMinus1Index);
  this->m_ls->InitializeSolution(m_SolutionTIndex);
  this->m_ls->InitializeSolution(m_TotalSolutionIndex);
  this->m_ls->InitializeSolution(m_SolutionTMinus1Index);
}

/**
 * Assemble the master stiffness matrix (also apply the MFCs to K)
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::AssembleKandM()
{
  // if no DOFs exist in a system, we have nothing to do
  if( this->m_NGFN <= 0 )
    {
    return;
    }

  Float lhsval;
  Float rhsval;
  this->m_NMFC = 0;  // number of MFC in a system

  // temporary storage for pointers to LoadBCMFC objects
  typedef std::vector<LoadBCMFC::Pointer> MFCArray;
  MFCArray mfcLoads;

  /*
   * Before we can start the assembly procedure, we need to know,
   * how many boundary conditions (MFCs) there are in a system.
   */
  mfcLoads.clear();

  int numLoads = this->m_FEMObject->GetLoadContainer()->Size();
  for( int l = 0; l < numLoads; l++ )
    {
    if( LoadBCMFC::Pointer l1 = dynamic_cast<LoadBCMFC *>( this->m_FEMObject->GetLoad(l).GetPointer() ) )
      {
      // store the index of an LoadBCMFC object for later
      l1->SetIndex(this->m_NMFC);
      mfcLoads.push_back(l1);
      // increase the number of MFC
      this->m_NMFC++;
      }
    }
  /**
   * Now we can assemble the master stiffness matrix
   * from element stiffness matrices
   */
  InitializeForSolution();

  /**
   * Step over all elements
   */
  int numElements = this->m_FEMObject->GetElementContainer()->Size();
  for( int e = 0; e < numElements;  e++ )
    {
    vnl_matrix<Float> Ke;
    this->m_FEMObject->GetElement(e)->GetStiffnessMatrix(Ke);  /*Copy the element stiffness matrix for
                                       faster access. */
    vnl_matrix<Float> Me;
    this->m_FEMObject->GetElement(e)->GetMassMatrix(Me);                      /*Copy the element mass
                                                      matrix for faster access.
                                                      */
    int Ne = this->m_FEMObject->GetElement(e)->GetNumberOfDegreesOfFreedom(); /*... same for element DOF
                                                      */

    Me = Me * m_Rho;
    /* step over all rows in in element matrix */
    for( int j = 0; j < Ne; j++ )
      {
      /* step over all columns in in element matrix */
      for( int k = 0; k < Ne; k++ )
        {
        /* error checking. all GFN should be =>0 and <NGFN */
        if( this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(j) >= this->m_NGFN
            || this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(k) >= this->m_NGFN )
          {
          throw FEMExceptionSolution(__FILE__, __LINE__, "SolverCrankNicolson::AssembleKandM()", "Illegal GFN!");
          }

        /* Here we finally update the corresponding element
         * in the master stiffness matrix. We first check if
         * element in Ke is zero, to prevent zeros from being
         * allocated in sparse matrix.
         */
        if( Ke(j, k) != Float(0.0) || Me(j, k) != Float(0.0) )
          {
          // left hand side matrix
          lhsval = ( Me(j, k) + m_Alpha * m_TimeStep * Ke(j, k) );
          this->m_ls->AddMatrixValue( this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(j),
                                      this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(k),
                                      lhsval, m_SumMatrixIndex );
          // right hand side matrix
          rhsval = ( Me(j, k) - ( 1. - m_Alpha ) * m_TimeStep * Ke(j, k) );
          this->m_ls->AddMatrixValue( this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(j),
                                      this->m_FEMObject->GetElement(e)->GetDegreeOfFreedom(k),
                                      rhsval, m_DifferenceMatrixIndex );
          }
        }
      }
    }
  /**
   * Step over all the loads to add the landmark contributions to the
   * appropriate place in the stiffness matrix
   */
  // int numLoads = m_FEMObject->GetLoadContainer()->Size();
  for( int l2 = 0; l2 < numLoads; l2++ )
    {
    if( LoadLandmark::Pointer l3 = dynamic_cast<LoadLandmark *>( this->m_FEMObject->GetLoad(l2).GetPointer() ) )
      {
      Element::ConstPointer ep = (l3->GetElementArray()[0]);
      Element::MatrixType   Le;
      ep->GetLandmarkContributionMatrix(l3->GetEta(), Le);
      int Ne = ep->GetNumberOfDegreesOfFreedom();
      // step over all rows in element matrix
      for( int j = 0; j < Ne; j++ )
        {
        // step over all columns in element matrix
        for( int k = 0; k < Ne; k++ )
          {
          // error checking, all GFN should be >=0 and < NGFN
          if( ep->GetDegreeOfFreedom(j) >= this->m_NGFN
              || ep->GetDegreeOfFreedom(k) >= this->m_NGFN )
            {
            throw FEMExceptionSolution(__FILE__, __LINE__, "SolverCrankNicolson::AssembleKandM()", "Illegal GFN!");
            }

          // Now update the corresponding element in the master
          // stiffness matrix and omit the zeros for the sparseness
          if( Le(j, k) != Float(0.0) )
            {
            // lhs matrix
            lhsval = m_Alpha * m_TimeStep * Le(j, k);
            this->m_ls->AddMatrixValue(ep->GetDegreeOfFreedom(j),
                                       ep->GetDegreeOfFreedom(k),
                                       lhsval, m_SumMatrixIndex);
            // rhs matrix
            rhsval = ( 1. - m_Alpha ) * m_TimeStep * Le(j, k);
            this->m_ls->AddMatrixValue(ep->GetDegreeOfFreedom(j),
                                       ep->GetDegreeOfFreedom(k),
                                       rhsval, m_DifferenceMatrixIndex);
            }
          }
        }
      }
    }

  /* step over all types of BCs */
  this->ApplyBC();  // BUG  -- are BCs applied appropriately to the problem?
}

/**
 * Assemble the master force vector
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::AssembleFforTimeStep(int dim)
{
  /* if no DOFs exist in a system, we have nothing to do */
  if( this->m_NGFN <= 0 )
    {
    return;
    }

  this->AssembleF(dim); // assuming assemblef uses index 0 in vector!

  typedef std::map<Element::DegreeOfFreedomIDType, Float> BCTermType;
  BCTermType bcterm;

  int numLoads = this->m_FEMObject->GetLoadContainer()->Size();
  for( int l2 = 0; l2 < numLoads; l2++ )
    {
    if( LoadBC::Pointer l1 = dynamic_cast<LoadBC *>( this->m_FEMObject->GetLoad(l2).GetPointer()  ) )
      {
      bcterm[l1->GetElement()->GetDegreeOfFreedom( l1->GetDegreeOfFreedom() )] =
        l1->GetValue()[dim];
      }
    } // end for LoadArray::iterator l
  // Now set the solution t_minus1 vector to fit the BCs
  for( BCTermType::iterator q = bcterm.begin(); q != bcterm.end(); q++ )
    {
    this->m_ls->SetVectorValue(q->first, 0.0, m_SolutionVectorTMinus1Index); // FIXME?
    this->m_ls->SetSolutionValue(q->first, 0.0, m_SolutionTMinus1Index);     // FIXME?
    this->m_ls->SetSolutionValue(q->first, 0.0, m_TotalSolutionIndex);
    }

  this->m_ls->MultiplyMatrixVector(m_DiffMatrixBySolutionTMinus1Index,
                                   m_DifferenceMatrixIndex, m_SolutionVectorTMinus1Index);
  for( unsigned int index = 0; index < this->m_NGFN; index++ )
    {
    RecomputeForceVector(index);
    }
  // Now set the solution and force vector to fit the BCs
  for( BCTermType::iterator q = bcterm.begin(); q != bcterm.end(); q++ )
    {
    this->m_ls->SetVectorValue(q->first, q->second, m_ForceTIndex);
    }
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::RecomputeForceVector(unsigned int index)
{     //
  Float ft   = this->m_ls->GetVectorValue(index, m_ForceTIndex);
  Float ftm1 = this->m_ls->GetVectorValue(index, m_ForceTMinus1Index);
  Float utm1 = this->m_ls->GetVectorValue(index, m_DiffMatrixBySolutionTMinus1Index);
  Float f = m_TimeStep * ( m_Alpha * ft + ( 1. - m_Alpha ) * ftm1 ) + utm1;

  this->m_ls->SetVectorValue(index, f, m_ForceTIndex);
}

// ----------------------------------------------------------------------------
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::GenerateData()
{
  /* Call Solver */
  this->RunSolver();
}

/**
 * Solve for the displacement vector u
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::RunSolver()
{
  if( m_Iterations == 0 )
    {
    if( m_UseMassMatrix )
      {
      this->AssembleKandM();
      }
    else
      {
      this->InitializeForSolution();
      this->AssembleK();
      }
    }

  if( m_UseMassMatrix )
    {
    this->AssembleFforTimeStep();
    }
  else
    {
    this->AssembleF();
    }

  /* FIXME - must verify that this is correct use of wrapper */
  /* FIXME Initialize the solution vector */
  this->m_ls->InitializeSolution(m_SolutionTIndex);
  this->m_ls->Solve();

  m_Iterations++;
  // call this externally    AddToDisplacements();
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::FindBracketingTriplet(Float *a, Float *b, Float *c)
{
  // in 1-D domain, we want to find a < b < c , s.t.  f(b) < f(a) && f(b) < f(c)
  //  see Numerical Recipes

  Float Gold = 1.618034;
  Float Glimit = 100.0;
  Float Tiny = 1.e-20;

  Float ax, bx, cx;

  ax = 0.0; bx = 1.;
  Float fc;
  Float fa = vcl_fabs( EvaluateResidual(ax) );
  Float fb = vcl_fabs( EvaluateResidual(bx) );

  Float ulim, u, r, q, fu, dum;

  if( fb > fa )
    {
    dum = ax; ax = bx; bx = dum;
    dum = fb; fb = fa; fa = dum;
    }

  cx = bx + Gold * ( bx - ax );  // first guess for c - the 3rd pt needed to
                                 // bracket the min
  fc = vcl_fabs( EvaluateResidual(cx) );

  while( fb > fc /*&& vcl_fabs(ax) < 3. && vcl_fabs(bx) < 3. && vcl_fabs(cx) <
                    3.*/)
    {
    r = ( bx - ax ) * ( fb - fc );
    q = ( bx - cx ) * ( fb - fa );
    Float denom = ( 2.0 * GSSign(GSMax(vcl_fabs(q - r), Tiny), q - r) );
    u = ( bx ) - ( ( bx - cx ) * q - ( bx - ax ) * r ) / denom;
    ulim = bx + Glimit * ( cx - bx );
    if( ( bx - u ) * ( u - cx ) > 0.0 )
      {
      fu = vcl_fabs( EvaluateResidual(u) );
      if( fu < fc )
        {
        ax = bx;
        bx = u;
        *a = ax; *b = bx; *c = cx;
        return;
        }
      else if( fu > fb )
        {
        cx = u;
        *a = ax; *b = bx; *c = cx;
        return;
        }

      u = cx + Gold * ( cx - bx );
      fu = vcl_fabs( EvaluateResidual(u) );
      }
    else if( ( cx - u ) * ( u - ulim ) > 0.0 )
      {
      fu = vcl_fabs( EvaluateResidual(u) );
      if( fu < fc )
        {
        bx = cx; cx = u; u = cx + Gold * ( cx - bx );
        fb = fc; fc = fu; fu = vcl_fabs( EvaluateResidual(u) );
        }
      }
    else if( ( u - ulim ) * ( ulim - cx ) >= 0.0 )
      {
      u = ulim;
      fu = vcl_fabs( EvaluateResidual(u) );
      }
    else
      {
      u = cx + Gold * ( cx - bx );
      fu = vcl_fabs( EvaluateResidual(u) );
      }

    ax = bx; bx = cx; cx = u;
    fa = fb; fb = fc; fc = fu;
    }

  if( vcl_fabs(ax) > 1.e3  || vcl_fabs(bx) > 1.e3 || vcl_fabs(cx) > 1.e3 )
    {
    ax = -2.0;  bx = 1.0;  cx = 2.0;
    } // to avoid crazy numbers caused by bad bracket (u goes nuts)

  *a = ax; *b = bx; *c = cx;
}

template <unsigned int VDimension>
Element::Float
SolverCrankNicolson<VDimension>
::BrentsMethod(Float tol, unsigned int MaxIters)
{
  // We should now have a, b and c, as well as f(a), f(b), f(c),
  // where b gives the minimum energy position;

  Float CGOLD = 0.3819660;
  Float ZEPS = 1.e-10;

  Float ax = 0.0, bx = 1.0, cx = 2.0;

  FindBracketingTriplet(&ax, &bx, &cx);

  Float xmin;

  unsigned int iter;

  Float a, b, d = 0., etemp, fu, fv, fw, fx, p, q, r, tol1, tol2, u, v, w, x, xm;

  Float e = 0.0;  // the distance moved on the step before last;

  a = ( ( ax  < cx ) ? ax : cx );
  b = ( ( ax  > cx ) ? ax : cx );

  x = w = v = bx;
  fw = fv = fx = vcl_fabs( EvaluateResidual(x) );
  for( iter = 1; iter <= MaxIters; iter++ )
    {
    xm = 0.5 * ( a + b );
    tol2 = 2.0 * ( tol1 = tol * vcl_fabs(x) + ZEPS );
    if( vcl_fabs(x - xm) <= ( tol2 - 0.5 * ( b - a ) ) )
      {
      xmin = x;
      SetEnergyToMin(xmin);
      return fx;
      }

    if( vcl_fabs(e) > tol1 )
      {
      r = ( x - w ) * ( fx - fv );
      q = ( x - v ) * ( fx - fw );
      p = ( x - v ) * q - ( x - w ) * r;
      q = 2.0 * ( q - r );
      if( q > 0.0 )
        {
        p = -1. * p;
        }
      q = vcl_fabs(q);
      etemp = e;
      e = d;
      if( vcl_fabs(p) >= vcl_fabs(0.5 * q * etemp) || p <= q * ( a - x ) || p >= q * ( b - x ) )
        {
        d = CGOLD * ( e = ( x >= xm ? a - x : b - x ) );
        }
      else
        {
        if( q == 0.0 )
          {
          q = q + ZEPS;
          }
        d = p / q;
        u = x + d;
        if( u - a < tol2 || b - u < tol2 )
          {
          d = GSSign(tol1, xm - x);
          }
        }
      }
    else
      {
      d = CGOLD * ( e = ( x >= xm ? a - x : b - x ) );
      }

    u = ( vcl_fabs(d) >= tol1 ? x + d : x + GSSign(tol1, d) );
    fu = vcl_fabs( EvaluateResidual(u) );
    if( fu <= fx )
      {
      if( u >= x )
        {
        a = x;
        }
      else
        {
        b = x;
        }
      v = w; w = x; x = u;
      fv = fw; fw = fx; fx = fu;
      }
    else
      {
      if( u < x )
        {
        a = u;
        }
      else
        {
        b = u;
        }
      if( fu <= fw || w == x )
        {
        v = w;
        w = u;
        fv = fw;
        fw = fu;
        }
      else if( fu <= fv || v == x || v == w )
        {
        v = u;
        fv = fu;
        }
      }
    }
  xmin = x;
  SetEnergyToMin(xmin);
  return fx;
}

template <unsigned int VDimension>
Element::Float
SolverCrankNicolson<VDimension>
::GoldenSection(Float tol, unsigned int MaxIters)
{
  // We should now have a, b and c, as well as f(a), f(b), f(c),
  // where b gives the minimum energy position;

  Float ax, bx, cx;

  FindBracketingTriplet(&ax, &bx, &cx);
  Float xmin, fmin;

  Float f1, f2, x0, x1, x2, x3;

  Float R = 0.6180339;
  Float C = ( 1.0 - R );

  x0 = ax;
  x3 = cx;
  if( vcl_fabs(cx - bx) > vcl_fabs(bx - ax) )
    {
    x1 = bx;
    x2 = bx + C * ( cx - bx );
    }
  else
    {
    x2 = bx;
    x1 = bx - C * ( bx - ax );
    }
  f1 = vcl_fabs( EvaluateResidual(x1) );
  f2 = vcl_fabs( EvaluateResidual(x2) );
  unsigned int iters = 0;
  while( vcl_fabs(x3 - x0) > tol * ( vcl_fabs(x1) + vcl_fabs(x2) ) && iters < MaxIters )
    {
    iters++;
    if( f2 < f1 )
      {
      x0 = x1; x1 = x2; x2 = R * x1 + C * x3;
      f1 = f2; f2 = vcl_fabs( EvaluateResidual(x2) );
      }
    else
      {
      x3 = x2; x2 = x1; x1 = R * x2 + C * x0;
      f2 = f1; f1 = vcl_fabs( EvaluateResidual(x1) );
      }
    }

  if( f1 < f2 )
    {
    xmin = x1;
    fmin = f1;
    }
  else
    {
    xmin = x2;
    fmin = f2;
    }

  SetEnergyToMin(xmin);
  return fmin;
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::SetEnergyToMin(Float xmin)
{
  for( unsigned int j = 0; j < this->m_NGFN; j++ )
    {
    Float SolVal;
    Float FVal;
#ifdef LOCE
    SolVal = xmin * this->m_ls->GetSolutionValue(j, m_SolutionTIndex)
      + ( 1. - xmin ) * this->m_ls->GetSolutionValue(j, m_SolutionTMinus1Index);

    FVal = xmin * this->m_ls->GetVectorValue(j, m_ForceTIndex)
      + ( 1. - xmin ) * this->m_ls->GetVectorValue(j, m_ForceTMinus1Index);
#endif
#ifdef TOTE
    SolVal = xmin * this->m_ls->GetSolutionValue(j, m_SolutionTIndex); // FOR TOT E
    FVal = xmin * this->m_ls->GetVectorValue(j, m_ForceTIndex);
#endif
    this->m_ls->SetSolutionValue(j, SolVal, m_SolutionTIndex);
    this->m_ls->SetVectorValue(j, FVal, m_ForceTIndex);
    }
}

template <unsigned int VDimension>
Element::Float
SolverCrankNicolson<VDimension>
::GetDeformationEnergy(Float t)
{
  Float DeformationEnergy = 0.0;
  Float iSolVal, jSolVal;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
// forming  U^T F
#ifdef LOCE
    iSolVal = t * ( this->m_ls->GetSolutionValue(i, m_SolutionTIndex) )
      + ( 1. - t ) * this->m_ls->GetSolutionValue(i, m_SolutionTMinus1Index);
#endif
#ifdef TOTE
    iSolVal = t * ( this->m_ls->GetSolutionValue(i, m_SolutionTIndex) );
#endif
// forming U^T K U
    Float TempRowVal = 0.0;
    for( unsigned int j = 0; j < this->m_NGFN; j++ )
      {
#ifdef LOCE
      jSolVal = t * ( this->m_ls->GetSolutionValue(j, m_SolutionTIndex) )
        + ( 1. - t ) * this->m_ls->GetSolutionValue(j, m_SolutionTMinus1Index);
#endif
#ifdef TOTE
      jSolVal = t * ( this->m_ls->GetSolutionValue(j, m_SolutionTIndex) )
        + this->m_ls->GetSolutionValue(j, m_TotalSolutionIndex);         // FOR TOT E
#endif
      TempRowVal += this->m_ls->GetMatrixValue(i, j, m_SumMatrixIndex) * jSolVal;
      }
    DeformationEnergy += iSolVal * TempRowVal;
    }
  return DeformationEnergy;
}

template <unsigned int VDimension>
Element::Float
SolverCrankNicolson<VDimension>
::EvaluateResidual(Float t)
{
  Float ForceEnergy = 0.0, FVal = 0.0;
  Float DeformationEnergy = 0.0;
  Float iSolVal, jSolVal;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
// forming  U^T F
#ifdef LOCE
    iSolVal = t * ( this->m_ls->GetSolutionValue(i, m_SolutionTIndex) )
      + ( 1. - t ) * this->m_ls->GetSolutionValue(i, m_SolutionTMinus1Index);
    FVal = this->m_ls->GetVectorValue(i, m_ForceTIndex);
    FVal = t * FVal + ( 1. - t ) * this->m_ls->GetVectorValue(i, m_ForceTMinus1Index);

    ForceEnergy += iSolVal * FVal;
#endif
#ifdef TOTE
    FVal = FVal + 0.0;
    iSolVal = t * ( this->m_ls->GetSolutionValue(i, m_SolutionTIndex) )
      + this->m_ls->GetSolutionValue(i, m_TotalSolutionIndex);         // FOR TOT E
    ForceEnergy += iSolVal * ( this->m_ls->GetVectorValue(i, m_ForceTotalIndex)
                               + t * this->m_ls->GetVectorValue(i, m_ForceTIndex) ); //
    // FOR
    // TOT
    // E
#endif
// forming U^T K U
    Float TempRowVal = 0.0;
    for( unsigned int j = 0; j < this->m_NGFN; j++ )
      {
#ifdef LOCE
      jSolVal = t * ( this->m_ls->GetSolutionValue(j, m_SolutionTIndex) )
        + ( 1. - t ) * this->m_ls->GetSolutionValue(j, m_SolutionTMinus1Index);
#endif
#ifdef TOTE
      jSolVal = t * ( this->m_ls->GetSolutionValue(j, m_SolutionTIndex) )
        + this->m_ls->GetSolutionValue(j, m_TotalSolutionIndex);         // FOR TOT E
#endif
      TempRowVal += this->m_ls->GetMatrixValue(i, j, m_SumMatrixIndex) * jSolVal;
      }
    DeformationEnergy += iSolVal * TempRowVal;
    }
  Float Energy = (Float)vcl_fabs(DeformationEnergy - ForceEnergy);
  return Energy;
}

/**
 * Copy solution vector u to the corresponding nodal values, which are
 * stored in node objects). This is standard post processing of the solution.
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::AddToDisplacements(Float optimum)
{
  /**
   * Copy the resulting displacements from
   * solution vector back to node objects.
   */
  Float maxs = 0.0, CurrentTotSolution, CurrentSolution, CurrentForce;
  Float mins2 = 0.0, maxs2 = 0.0;
  Float absmax = 0.0;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
#ifdef TOTE
    CurrentSolution = this->m_ls->GetSolutionValue(i, m_SolutionTIndex);
#endif
    if( CurrentSolution < mins2 )
      {
      mins2 = CurrentSolution;
      }
    else if( CurrentSolution > maxs2 )
      {
      maxs2 = CurrentSolution;
      }
    if( vcl_fabs(CurrentSolution) > absmax )
      {
      absmax = vcl_fabs(CurrentSolution);
      }

//  note: set rather than add - i.e. last solution of system not total solution
#ifdef LOCE
    CurrentSolution = optimum * this->m_ls->GetSolutionValue(i, m_SolutionTIndex)
      + ( 1. - optimum ) * this->m_ls->GetVectorValue(i, m_SolutionVectorTMinus1Index);
    CurrentForce = optimum * this->m_ls->GetVectorValue(i, m_ForceTIndex)
      + ( 1. - optimum ) * this->m_ls->GetVectorValue(i, m_ForceTMinus1Index);
    this->m_ls->SetVectorValue(i, CurrentSolution, m_SolutionVectorTMinus1Index);
    this->m_ls->SetSolutionValue(i, CurrentSolution, m_SolutionTMinus1Index);
    this->m_ls->SetVectorValue(i, CurrentForce, m_ForceTMinus1Index);  // now set t
    // minus one
    // force vector
    // correctly
#endif
#ifdef TOTE
    CurrentSolution = optimum * CurrentSolution;
    CurrentForce = optimum * this->m_ls->GetVectorValue(i, m_ForceTIndex);
    this->m_ls->SetVectorValue(i, CurrentSolution, m_SolutionVectorTMinus1Index); // FOR
    // TOT
    // E
    this->m_ls->SetSolutionValue(i, CurrentSolution, m_SolutionTMinus1Index);     // FOR
    // TOT
    // E
    this->m_ls->SetVectorValue(i, CurrentForce, m_ForceTMinus1Index);
#endif

    this->m_ls->AddSolutionValue(i, CurrentSolution, m_TotalSolutionIndex);
    this->m_ls->AddVectorValue(i, CurrentForce, m_ForceTotalIndex);
    CurrentTotSolution = this->m_ls->GetSolutionValue(i, m_TotalSolutionIndex);

    if( vcl_fabs(CurrentTotSolution) > maxs )
      {
      maxs = vcl_fabs(CurrentTotSolution);
      }
    }

  m_CurrentMaxSolution = absmax;
}

/**
 * Compute maximum and minimum solution values.
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::PrintMinMaxOfSolution()
{
  /**
   * Copy the resulting displacements from
   * solution vector back to node objects.
   */
  Float mins = 0.0, maxs = 0.0;
  Float mins2 = 0.0, maxs2 = 0.0;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
    Float CurrentSolution = this->m_ls->GetSolutionValue(i, m_SolutionTIndex);
    if( CurrentSolution < mins2 )
      {
      mins2 = CurrentSolution;
      }
    else if( CurrentSolution > maxs2 )
      {
      maxs2 = CurrentSolution;
      }
    CurrentSolution = this->m_ls->GetSolutionValue(i, m_TotalSolutionIndex);
    if( CurrentSolution < mins )
      {
      mins = CurrentSolution;
      }
    else if( CurrentSolution > maxs )
      {
      maxs = CurrentSolution;
      }
    }
}

/**
 * Copy solution vector u to the corresponding nodal values, which are
 * stored in node objects). This is standard post processing of the solution.
 */
template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::AverageLastTwoDisplacements(Float t)
{
  Float maxs = 0.0;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
    Float temp = this->m_ls->GetSolutionValue(i, m_SolutionTIndex);
    Float temp2 = this->m_ls->GetSolutionValue(i, m_SolutionTMinus1Index);
    Float newsol = t * ( temp ) + ( 1. - t ) * temp2;
    this->m_ls->SetSolutionValue(i, newsol, m_SolutionTMinus1Index);
    this->m_ls->SetVectorValue(i, newsol, m_SolutionVectorTMinus1Index);
    this->m_ls->SetSolutionValue(i, newsol, m_SolutionTIndex);
    if( newsol > maxs )
      {
      maxs = newsol;
      }
    }
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::ZeroVector(int which)
{
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
    this->m_ls->SetVectorValue(i, 0.0, which);
    }
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::PrintDisplacements()
{
  std::cout <<  " printing current displacements " << std::endl;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
    std::cout << this->m_ls->GetSolutionValue(i, m_TotalSolutionIndex) << std::endl;
    }
}

template <unsigned int VDimension>
void
SolverCrankNicolson<VDimension>
::PrintForce()
{
  std::cout <<  " printing current forces " << std::endl;
  for( unsigned int i = 0; i < this->m_NGFN; i++ )
    {
    std::cout << this->m_ls->GetVectorValue(i, m_ForceTIndex) << std::endl;
    }
}

}
}  // end namespace itk::fem
#endif // __itkFEMSolverCrankNicolson_hxx